Determination of Magnetic Symmetries by Convergent Beam Electron Diffraction

TL;DR

The paper presents magnetic convergent-beam electron diffraction (CBED) as a TEM-based method to determine magnetic point-group symmetries with nanometer-scale spatial resolution. By formulating the paraxial scattering framework and systematically constructing 125 magnetic CBED groups, the authors map these groups to the 122 magnetic point groups, enabling symmetry extraction from multiple slab orientations. They validate the approach with multislice simulations on antiferromagnets (e.g., LaMnAsO and NiO) and discuss practical experimental considerations, including signal strength, thermal diffuse scattering, and potential zero-field TEM setups. The work promises a high-resolution alternative to neutron diffraction for magnetic-symmetry determination and offers guidelines for mapping symmetry across domains and defects in magnetic materials.

Abstract

Convergent-beam electron diffraction (CBED) is a well-established probe for spatial symmetries of crystalline samples, mainly exploiting the well-defined mapping between the diffraction groups (symmetry group of CBED patterns) and the point-group symmetries of the crystalline sample. In this work, we extend CBED to determine magnetic point groups. We construct all magnetic CBED groups, of which there exist 125. Then, we provide the complete mapping of the 122 magnetic point groups to corresponding magnetic CBED groups for all crystal orientations. In order to verify the group-theoretical considerations, we conduct electron-scattering simulations on antiferromagnetic crystals and provide guidelines for the experimental realization. Based on its feasibility using existing technology, as well as on its accuracy, high spatial resolution, and small required sample size, magnetic CBED promises to be become a valuable alternative method for magnetic structure determination.

Figures (9)

• Figure 1: Minimal optical model of a CBED setup. The beam converges in the sample plane, while the beam-limiting aperture is in the front focal plane of the condenser lens. The CBED pattern is recorded in the back focal plane (far field), where diffracted beams appear as disks because of the convergent illumination. Sample space as well as focal plane (far field) coordinates are indicated.
• Figure 2: Magnetic point group $4'/m$ symmetry broken by the slab geometry of a thin TEM sample of tetragonal symmetry. "'DG'" stands for diffraction group, i.e., magnetic CBED group.
• Figure 3: Crystal structure of LaMnAsO with directions of the Mn magnetic moments.
• Figure 4: CBED patterns pertaining to a LaMnAsO slab (thickness $\approx 100\,\mathrm{nm}$) in $\left[100\right]$ orientation. Reference nonmagnetic and magnetic CBED patterns are shown at beginning of first and third row. Otherwise, differences of CBED patterns are depicted. Rows indicate (1) nonmagnetic, (2) $z$-reversed nonmagnetic, (3) magnetic, (4) time- (magnetization-) reversed magnetic, (5) $z$-reversed magnetic, (6) time- and $z$-reversed magnetic structure. Columns indicate (1) as computed, (2) $x-$mirrored, (3) $y-$mirrored, and (4) $180^\circ$-rotated CBED patterns, see text.
• Figure 5: Crystal structure of NiO with directions of the magnetic moments. Note that only one structural cubic unit cell is displayed. The cubic antiferromagnetic unit cell is twice as large in each direction. The magnetic monoclinic unit cell, on the other hand, lies askew within the cubic cell and has the same volume.